Fundamental theorem of linear algebra
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In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m×n matrix A and its LDU factorization:
wherein P is a permutation matrix, L is a lower triangular matrix, D is a diagonal matrix, and U is an upper triangular matrix. At a more abstract level there is an interpretation that reads it in terms of a linear mapping and its transpose.
First, each matrix A induces four fundamental subspaces. These fundamental subspaces are:
| name of subspace | definition | containing space | dimension | basis |
|---|---|---|---|---|
| column space, range or image | im(A) or range(A) |  |
r (rank) | The r columns corresponding to those with pivots in  |
| nullspace or kernel | ker(A) or null(A) |  |
n − r (nullity) | The (n − r) columns of x in the solution of  |
| row space or coimage | im(AT) or range(AT) | |
r | The r rows corresponding to those with pivots in |
| left nullspace or cokernel | ker(AT) or null(AT) | |
m − r | The last (m − r) rows of  |
Secondly:
- In ,
, that is, the nullspace is the orthogonal complement of the row space - In ,
, that is, the left nullspace is the orthogonal complement of the column space.
The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.
Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as 
and 
: the kernel and image of A * are the cokernel and coimage of A.
[edit] See also
[edit] References
- Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
- Strang, Gilbert (1993), "The fundamental theorem of linear algebra", American Mathematical Monthly 100 (9): 848–855, http://www.eng.iastate.edu/~julied/classes/CE570/Notes/strangpaper.pdf
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